Generic measures with slowly decaying Fourier coefficients

TL;DR

This paper explores threshold phenomena in weighted $ ext{ell}^2$-spaces, characterizing critical regimes for constructing elements with small support or poor range, using sparse Fourier spectrum techniques and demonstrating their genericity.

Contribution

It introduces a novel analysis of threshold phenomena in weighted $ ext{ell}^2$-spaces with sparse Fourier spectrum methods and shows the genericity of these constructions.

Findings

Optimal characterization of critical regimes

Construction of elements with small support or bad range

Demonstration of genericity of these constructions

Abstract

We investigate threshold phenomena in weighted $\ell^2$-spaces and characterize the critical regimes where elements with either small support or maximally bad range can be constructed. Our results are shown to be optimal in several respects, and our proofs principally rely on techniques involving sparse Fourier spectrum. We further show that these seemingly pathological constructions are actually generic from certain categorical perspectives.